you and a companion are driving a twisty stretch of road in a car with a speedometer but no odometer. to…
you and a companion are driving a twisty stretch of road in a car with a speedometer but no odometer. to find out how long this road is, you record the cars velocity at 10 - second intervals. a. estimate the length of the road using left - endpoint values. ft
Answer
Explanation:
Step1: Recall left - endpoint Riemann sum formula
The left - endpoint Riemann sum for approximating the distance $d$ (where velocity $v(t)$ is given) over an interval $[a,b]$ with $n$ sub - intervals of width $\Delta t$ is $d\approx\sum_{i = 0}^{n - 1}v(t_i)\Delta t$. Here, $\Delta t=10$ s.
Step2: Identify the number of sub - intervals and left - endpoint velocities
We have data from $t = 0$ to $t=120$ s with $\Delta t = 10$ s, so $n = 12$. The left - endpoint velocities are $v(0)=0$, $v(10)=20$, $v(20)=14$, $v(30)=15$, $v(40)=43$, $v(50)=33$, $v(60)=13$, $v(70)=22$, $v(80)=21$, $v(90)=30$, $v(100)=28$, $v(110)=46$.
Step3: Calculate the sum
[ \begin{align*} d&\approx10\times(0 + 20+14 + 15+43+33+13+22+21+30+28+46)\ &=10\times(275)\ &=2750 \end{align*} ]
Answer:
$2750$